Higher-order scheme-independent series expansions of ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR}}$ and ${\beta }_{\mathrm{IR}}^{\prime }$ in conformal field theories

We study a vectorial asymptotically free gauge theory, with gauge group $G$ and $N_f$ massless fermions in a representation $R$ of this group, that exhibits an infrared (IR) zero in its beta function, $\beta$, at the coupling $\alpha ={\alpha }_{\mathrm{IR}}$ in the non-Abelian Coulomb phase. For general $G$ and $R$, we calculate the scheme-independent series expansions of (i) the anomalous dimension of the fermion bilinear, ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR}}$, to $O({\mathrm{\Delta }}_f^4)$ and (ii) the derivative ${\beta }^{\prime }=d\beta /d\alpha$, to $O({\mathrm{\Delta }}_f^5)$, both evaluated at ${\alpha }_{\mathrm{IR}}$, where ${\mathrm{\Delta }}_f$ is an $N_f$-dependent expansion variable. These are the highest orders to which these expansions have been calculated. We apply these general results to theories with $G=\mathrm{SU}(N_c)$ and $R$ equal to the fundamental, adjoint, and symmetric and antisymmetric rank-2 tensor representations. It is shown that for all of these representations, ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR}}$, calculated to the order ${\mathrm{\Delta }}_f^p$, with $1\le p\le 4$, increases monotonically with decreasing $N_f$ and, for fixed $N_f$, is a monotonically increasing function of $p$. Comparisons of our scheme-independent calculations of ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR}}$ and ${\beta }_{\mathrm{IR}}^{\prime }$ are made with our earlier higher $n$-loop values of these quantities, and with lattice measurements. For $R=F$, we present results for the limit $N_c\to \infty$ and $N_f\to \infty$ with $N_f/N_c$ fixed. We also present expansions for ${\alpha }_{\mathrm{IR}}$ calculated to $O({\mathrm{\Delta }}_f^4)$.

Figure 1

Plot of ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR},F,{\mathrm{\Delta }}_f^p}$ (labeled as ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR}}$ on the vertical axis in this and subsequent graphs) for $N_c=2$, i.e., $G=\mathrm{SU}(2)$, and $1\le p\le 4$ as a function of $N_f\in I_{\mathrm{IRZ}}$. From bottom to top, the curves (with colors online) refer to ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR},F,{\mathrm{\Delta }}_f}$ (red), ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR},F,{\mathrm{\Delta }}_f^2}$ (green), ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR},F,{\mathrm{\Delta }}_f^3}$ (blue), and ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR},F,{\mathrm{\Delta }}_f^4}$ (black).

Figure 2

Plot of ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR},F,{\mathrm{\Delta }}_f^p}$ for $N_c=4$ and $1\le p\le 4$ as a function of $N_f\in I_{\mathrm{IRZ}}$. From bottom to top, the curves (with colors online) refer to ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR},F,{\mathrm{\Delta }}_f}$ (red), ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR},F,{\mathrm{\Delta }}_f^2}$ (green), ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR},F,{\mathrm{\Delta }}_f^3}$ (blue), and ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR},F,{\mathrm{\Delta }}_f^4}$ (black).

Figure 3

Plot of ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR},F,{\mathrm{\Delta }}_f^p}$ for $N_c=3$ and $1\le p\le 4$ as a function of $N_f\in I_{\mathrm{IRZ}}$. From bottom to top, the curves (with colors online) refer to ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR},F,{\mathrm{\Delta }}_f}$ (red), ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR},F,{\mathrm{\Delta }}_f^2}$ (green), ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR},F,{\mathrm{\Delta }}_f^3}$ (blue), and ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR},F,{\mathrm{\Delta }}_f^4}$ (black).

Figure 4

Plot of ${\gamma }_{\mathrm{IR},F,{\mathrm{\Delta }}_r^p}$ for $1\le p\le 4$ as a function of $r\in I_{\mathrm{IRZ},r}$ in the LNN limit (3.21). From bottom to top, the curves (with colors online) refer to ${\gamma }_{\mathrm{IR},F,{\mathrm{\Delta }}_r}$ (red), ${\gamma }_{\mathrm{IR},F,{\mathrm{\Delta }}_r^2}$ (green), ${\gamma }_{\mathrm{IR},F,{\mathrm{\Delta }}_r^3}$ (blue), and ${\gamma }_{\mathrm{IR},F,{\mathrm{\Delta }}_r^4}$ (black).

Figure 5

Plot of ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR},adj,{\mathrm{\Delta }}_f^p}$ for $G=\mathrm{SU}(2)$ and $1\le p\le 4$ as a function of $N_f\in I_{\mathrm{IRZ}}$ for $R=adj$ and $N_f=2$. From bottom to top, the curves (with colors online) refer to ${\gamma }_{\mathrm{IR},adj,{\mathrm{\Delta }}_f}$ (red), ${\gamma }_{\mathrm{IR},adj,{\mathrm{\Delta }}_f^2}$ (green), ${\gamma }_{\mathrm{IR},adj,{\mathrm{\Delta }}_f^3}$ (blue), and ${\gamma }_{\mathrm{IR},adj,{\mathrm{\Delta }}_f^4}$ (black).

Figure 6

Plot of ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR},S_2,{\mathrm{\Delta }}_f^p}$ for $N_c=3$ and $1\le p\le 4$ as a function of $N_f$. Here, $S_2$ denotes the symmetric rank-2 tensor representation. From bottom to top, the curves (with colors online) refer to ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR},S_2,{\mathrm{\Delta }}_f}$ (red), ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR},S_2,{\mathrm{\Delta }}_f^2}$ (green), ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR},S_2,{\mathrm{\Delta }}_f^3}$ (blue), and ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR},S_2,{\mathrm{\Delta }}_f^4}$ (black).

Figure 7

Plot of ${\beta }_{\mathrm{IR},F,{\mathrm{\Delta }}_f^p}^{\prime }$ (labeled as ${\beta }_{\mathrm{IR}}^{\prime }$ on the vertical axis) for $N_c=2$ and $2\le p\le 5$ as a function of $N_f\in I_{\mathrm{IRZ}}$. From bottom to top, the curves (with colors online) refer to ${\beta }_{\mathrm{IR},F,{\mathrm{\Delta }}_f^2}^{\prime }$ (red), ${\beta }_{\mathrm{IR},F,{\mathrm{\Delta }}_f^3}^{\prime }$ (green), ${\beta }_{\mathrm{IR},F,{\mathrm{\Delta }}_f^4}^{\prime }$ (blue), and ${\beta }_{\mathrm{IR},F,{\mathrm{\Delta }}_f^5}^{\prime }$ (black).

Figure 8

Plot of ${\beta }_{\mathrm{IR},F,{\mathrm{\Delta }}_f^p}^{\prime }$ for $N_c=3$ and $2\le p\le 5$ as a function of $N_f\in I_{\mathrm{IRZ}}$. From bottom to top, the curves (with colors online) refer to ${\beta }_{\mathrm{IR},F,{\mathrm{\Delta }}_f^2}^{\prime }$ (red), ${\beta }_{\mathrm{IR},F,{\mathrm{\Delta }}_f^3}^{\prime }$ (green), ${\beta }_{\mathrm{IR},F,{\mathrm{\Delta }}_f^4}^{\prime }$ (blue), and ${\beta }_{\mathrm{IR},F,{\mathrm{\Delta }}_f^5}^{\prime }$ (black).

Figure 9

Plot of ${\beta }_{\mathrm{IR},F,{\mathrm{\Delta }}_f^p}^{\prime }$ for $N_c=4$ and $2\le p\le 5$ as a function of $N_f\in I_{\mathrm{IRZ}}$. From bottom to top, the curves (with colors online) refer to ${\beta }_{\mathrm{IR},F,{\mathrm{\Delta }}_f^2}^{\prime }$ (red), ${\beta }_{\mathrm{IR},F,{\mathrm{\Delta }}_f^3}^{\prime }$ (green), ${\beta }_{\mathrm{IR},F,{\mathrm{\Delta }}_f^4}^{\prime }$ (blue), and ${\beta }_{\mathrm{IR},F,{\mathrm{\Delta }}_f^5}^{\prime }$ (black).

Figure 10

Plot of ${\alpha }_{\mathrm{IR},F,{\mathrm{\Delta }}_f^p}$ (denoted as ${\alpha }_{\mathrm{IR}}$ on the vertical axis) with $1\le p\le 4$ for $G=\mathrm{SU}(2)$, as functions of $N_f\in I_{\mathrm{IRZ}}$. From bottom to top, the curves (with colors online) refer to ${\alpha }_{\mathrm{IR},F,{\mathrm{\Delta }}_f}$ (red), ${\alpha }_{\mathrm{IR},F,{\mathrm{\Delta }}_f^2}$ (green), ${\alpha }_{\mathrm{IR},F,{\mathrm{\Delta }}_f^3}$ (blue), and ${\alpha }_{\mathrm{IR},F,{\mathrm{\Delta }}_f^4}$ (black). Note that the curves for ${\alpha }_{\mathrm{IR},F,{\mathrm{\Delta }}_f^3}$ and ${\alpha }_{\mathrm{IR},{\mathrm{\Delta }}_f^4}$ are so close as to be indistinguishable in this figure.

Figure 11

Plot of ${\alpha }_{\mathrm{IR},F,{\mathrm{\Delta }}_f^p}$ with $1\le p\le 4$ for $G=\mathrm{SU}(3)$, as functions of $N_f\in I_{\mathrm{IRZ}}$. From bottom to top, the curves (with colors online) refer to ${\alpha }_{\mathrm{IR},F,{\mathrm{\Delta }}_f}$ (red), ${\alpha }_{\mathrm{IR},F,{\mathrm{\Delta }}_f^2}$ (green), ${\alpha }_{\mathrm{IR},F,{\mathrm{\Delta }}_f^3}$ (blue), and ${\alpha }_{\mathrm{IR},F,{\mathrm{\Delta }}_f^4}$ (black).

Figure 12

Plot of ${\alpha }_{\mathrm{IR},F,{\mathrm{\Delta }}_f^p}$ with $1\le p\le 4$ for $G=\mathrm{SU}(4)$, as functions of $N_f\in I_{\mathrm{IRZ}}$. From bottom to top, the curves (with colors online) refer to ${\alpha }_{\mathrm{IR},F,{\mathrm{\Delta }}_f}$ (red), ${\alpha }_{\mathrm{IR},F,{\mathrm{\Delta }}_f^2}$ (green), ${\alpha }_{\mathrm{IR},F,{\mathrm{\Delta }}_f^3}$ (blue), and ${\alpha }_{\mathrm{IR},F,{\mathrm{\Delta }}_f^4}$ (black).